A robust Petrov-Galerkin discretisation of convection-diffusion equations. (English) Zbl 1364.65191
Summary: A Petrov-Galerkin discretisation is studied of an ultra-weak variational formulation of the convection-diffusion equation in a mixed form. To arrive at an implementable method, the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space. To prevent that this latter space has to be taken increasingly large for vanishing diffusion, a formulation is constructed that is well-posed in the limit case of a pure transport problem. Numerical experiments show approximations that are very close to the best approximations to the solution from the trial space, uniformly in the size of the diffusion term.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Barrett, J. W.; Morton, K. W., Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comput. Methods Appl. Mech. Engrg., 45, 1-3, 97-122 (1984) · Zbl 0562.76086 |
| [2] | Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations, 27, 1, 70-105 (2011) · Zbl 1208.65164 |
| [3] | Cohen, A.; Dahmen, W.; Welper, G., Adaptivity and variational stabilization for convection-diffusion equations, ESAIM Math. Model. Numer. Anal., 46, 1247-1273 (2012) · Zbl 1270.65065 |
| [4] | Fortin, M., An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér., 11, 4, 341-354 (1977), iii · Zbl 0373.65055 |
| [5] | Gopalakrishnan, J.; Qiu, W., An analysis of the practical DPG method, Math. Comp., 83, 286, 537-552 (2014) · Zbl 1282.65154 |
| [6] | Kato, T., Estimation of iterated matrices, with application to the von Neumann condition, Numer. Math., 2, 22-29 (1960) · Zbl 0119.32001 |
| [7] | Xu, J.; Zikatanov, L., Some observations on Babuška and Brezzi theories, Numer. Math., 94, 1, 195-202 (2003) · Zbl 1028.65115 |
| [8] | Dahmen, W.; Huang, C.; Schwab, Ch.; Welper, G., Adaptive Petrov-Galerkin methods for first order transport equations, SIAM J. Numer. Anal., 50, 5, 2420-2445 (2012) · Zbl 1260.65091 |
| [9] | Zitelli, J.; Muga, I.; Demkowicz, L.; Gopalakrishnan, J.; Pardo, D.; Calo, V. M., A class of discontinuous Petrov-Galerkin methods. Part IV: the optimal test norm and time-harmonic wave propagation in 1D, J. Comput. Phys., 230, 7, 2406-2432 (2011) · Zbl 1316.76054 |
| [10] | Bottasso, C. L.; Micheletti, S.; Sacco, R., The discontinuous Petrov-Galerkin method for elliptic problems, Comput. Methods Appl. Mech. Engrg., 191, 31, 3391-3409 (2002) · Zbl 1010.65050 |
| [11] | Bardos, C., Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. Éc. Norm. Supér. (4), 3, 185-233 (1970) · Zbl 0202.36903 |
| [12] | Lions, J.-L.; Magenes, E., (Non-Homogeneous Boundary Value Problems and Applications. Vol. I (1972), Springer-Verlag: Springer-Verlag New York), Translated from the French by P. Kenneth, Die Grundlehren der Mathematischen Wissenschaften, Band 181 · Zbl 0227.35001 |
| [13] | Cohen, A.; Echeverry, L. M.; Sun, Q., Finite element wavelets, Technical Report (2000), Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie |
| [14] | Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev., 34, 581-613 (1992) · Zbl 0788.65037 |
| [15] | Bramble, J. H.; Pasciak, J. E.; Vassilevski, P. S., Computational scales of Sobolev norms with application to preconditioning, Math. Comp., 69, 230, 463-480 (2000) · Zbl 0941.65052 |
| [16] | Cao, T., Hierarchical basis methods for hypersingular integral equations, IMA J. Numer. Anal., 17, 4, 603-619 (1997) · Zbl 0887.65137 |
| [17] | Broersen, D.; Stevenson, R. P., A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form, IMA J. Numer. Anal. (2014) |
| [18] | Brooks, A. N.; Hughes, Th. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 1-3, 199-259 (1982), FENOMECH’81, Part I (Stuttgart, 1981) · Zbl 0497.76041 |
| [19] | Egger, H.; Schöberl, J., A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems, IMA J. Numer. Anal., 30, 4, 1206-1234 (2010) · Zbl 1204.65133 |
| [20] | Demkowicz, L.; Heuer, N., Robust DPG method for convection-dominated diffusion problems, SIAM J. Numer. Anal., 51, 5, 2514-2537 (2013) · Zbl 1290.65088 |
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